Open-source finite-element tests of the Rosenzweig-Porter hypothesis for freely vibrating plates: preliminary results

The companion paper "Why Rosenzweig-Porter?" proposes that the intermediate (Rosenzweig-Porter, RP) spectral statistics observed in freely vibrating thin plates arise from evanescent-wave coupling at free edges acting on an integrable base, and it registers the numerical tests that would decide the hypothesis, assigning them to COMSOL "or finite elements". We report the first execution of that program with a fully open-source stack (scikit-fem, Argyris elements, certified eigensolvers), under preregistered readings and measurement-based accuracy gates. Main findings: (i) the boundary-controlled Poisson-to-intermediate transition is confirmed at 3.6 sigma; (ii) the geometry program yields a three-tier hierarchy governed by the boundary's adaptedness to separable coordinates — exact reduction (disk) gives Poisson; coordinate-adapted boundaries (ellipse, sector, rectangle) give weak sparse coupling; unadapted boundaries (triangle, superellipse p ≠ 2) give full coupling — with a +12.2 sigma step between the ellipse (p = 2) and the smooth soft-cornered superellipse (p = 3); (iii) Gap A, executed for the first time on the true operator, shows N-independent inverse participation ratios: at accessible sector sizes each free-plate mode is approximately 1-2 free-free beam products, i.e. sparse avoided-crossing hybridization rather than an RP-fractal phase. Extending the program beyond the original registry: the Levy-separable mixed configuration lands exactly on Poisson; both exactly-reduced controls (disk, annulus) are confirmed; the coupling is material-tunable and post-designable via a validated second-order control law; the Mindlin thickness sweep confirms the thick-plate prediction at +5.4 sigma; the sector reaches +3.9 sigma at 1600 modes, and a faithful replication of the published sector protocol — which the source reveals to be a pooled angle sweep — supports its direction at +9.5 sigma, with the pooled-sweep methodology validated as bias-free. The central Gap A question is resolved by a controlled protocol comparison: the registered truncated-operator ladder does develop RP-like fractal scaling (D2 = 0.42-0.50) — but a purpose-built C0 interior-penalty discretization shows the true-operator eigenvectors stay sparse through N = 1024 per sector at gate-certified windows (indicatively 2048): the scaling is a property of the truncation protocol, not of the plate. The same comparison on the unadapted tier finds a weak genuine true-operator delocalization (triangle D2 ≈ 0.14 at 4.8 sigma) that the truncated protocol exaggerates seven-fold; a refine-7 instrument confirms the third hierarchy point on the superellipse p = 10 (D2_true = 0.249 ± 0.072, 3.5 sigma), ordering the genuine effect exactly by boundary adaptedness (0.01 → 0.14 → 0.25) — Gap A closes protocol-negative across the whole tier hierarchy, with a real, ordered, weak multifractal residue in its place. Finally, an in-plane rotating-plate experiment delivers a mechanical GOE-to-GUE crossover together with its protection theorem: a chiral rotor lands exactly on GUE (0.5992 ± 0.0071) while the mirror-symmetric rotor is pinned to GOE at every speed by the surviving antiunitary sigma_v T, and asymmetric mistuning breaks that protection at +7.8 sigma; feasibility, prestress, and finite-deformation analyses single out soft (elastomer) rotors as the laboratory route — the crossover completes within validated moderate-strain physics at about 1,500 RPM for a decimeter silicone disk. A designed mechanical mixture-of-experts closes the constructive bridge to the sibling AI program: the fabricated fractal dimension is a monotone dial in coupler connectivity, reaching D2 = 0.63-0.69 at router-like (complete-graph) coupling, inside the AI-side architecture band. All experiments, gates, and results are reproducible from the public repository (https://github.com/arturoornelasb/plates-rp-fem); the supplementary package archives a snapshot of record.